nLab spectrification of a sequential spectrum type



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels



Stable Homotopy theory



In classical algebraic topology, there is a spectrification functor which is left adjoint to the inclusion of spectra in prespectra. For instance, this is how a suspension spectrum is constructed: by spectrifying the prespectrum X nΣ nAX_n \coloneqq \Sigma^n A. There is an analogue in homotopy type theory which constructs the spectrification of a sequential spectrum type (i.e. prespectrum types) into a Omega-spectrum type (i.e. spectrum types).


The following higher inductive type should construct spectrification in homotopy type theory (though this has not yet been verified formally). (There are some abuses of notation below, which can be made precise using Coq typeclasses and implicit arguments.)

Inductive spectrify (X : prespectrum) : nat -> Type :=
| to_spectrify : forall n, X n -> spectrify X n
| spectrify_glue : forall n, spectrify X n ->
    to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))
| to_spectrify_is_prespectrum_map : forall n (x : X n),
    spectrify_glue n (to_spectrify n x)
    == loop_functor (to_spectrify (S n)) (glue n x)
| spectrify_glue_retraction : forall n
    (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
    spectrify X n
| spectrify_glue_retraction_is_retraction : forall n (sx : spectrify X n),
    spectrify_glue_retraction n (spectrify_glue n sx) == sx
| spectrify_glue_section : forall n
    (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
    spectrify X n
| spectrify_glue_section_is_section : forall n
    (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
    spectrify_glue n (spectrify_glue_section n p) == p.

We can unravel this as follows, using more traditional notation. Let LXL X denote the spectrification being constructed. The first constructor says that each (LX) n(L X)_n comes with a map from X nX_n, called n\ell_n say (denoted to_spectrify n above). This induces a basepoint in each type (LX) n(L X)_n, namely the image n(*)\ell_n(*) of the basepoint of X nX_n. The many occurrences of

to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))

simply refer to the based loop space of Ω n+1(*)(LX) n+1\Omega_{\ell_{n+1}(*)} (L X)_{n+1} of (LX) n+1(L X)_{n+1} at this base point.

Thus, the second constructor spectrify_glue gives the structure maps (LX) nΩ(LX) n+1(L X)_n \to \Omega (L X)_{n+1} to make LXL X into a prespectrum. Similarly, the third constructor says that the maps n:X n(LX) n\ell_n\colon X_n \to (L X)_n commute with the structure maps up to a specified homotopy.

Since the basepoints of the types (LX) n(L X)_n are induced from those of each X nX_n, this automatically implies that the maps (LX) nΩ(LX) n+1(L X)_n \to \Omega (L X)_{n+1} are pointed maps (up to a specified homotopy) and that the n\ell_n commute with these pointings (up to a specified homotopy). This makes \ell into a map of prespectra.

Finally, the fourth through seventh constructors say that LXL X is a spectrum, by giving h-isomorphism data: a retraction and a section for each glue map (LX) nΩ(LX) n+1(L X)_n \to \Omega (L X)_{n+1}. We could use adjoint equivalence data as we did for localization, but this approach avoids the presence of level-3 path constructors. (We could have used h-iso data in localization too, thereby avoiding even level-2 constructors there.) It is important, in general, to use a sort of equivalence data which forms an h-prop; otherwise we would be adding structure rather than merely the property of such-and-such map being an equivalence.

See also


Last revised on June 9, 2022 at 05:58:13. See the history of this page for a list of all contributions to it.